Extrinsic semiconductor

What is an extrinsic semiconductor?

In plain words: pure silicon has few free carriers at room temperature; doping allows their density and dominant type to be controlled. Swap about one silicon atom in a million for an atom with one electron more, and that spare electron is almost free; swap it for an atom with one electron fewer, and a bond is left one electron short, a hole. Either way the number of carriers jumps, and you choose which kind.

Doping

• The conductivity of an intrinsic semiconductor depends on its temperature, but at room temperature it is very low: “no important electronic devices can be developed using these semiconductors”, so impurities are added to raise it (NCERT page 329). Strictly, doping also gives a controlled population of majority carriers (reasoned).
• Adding a few parts per million (ppm) of a suitable impurity increases the conductivity manifold (NCERT page 329).
• Suitable substitutional dopants replace host atoms without destroying the crystal. In NCERT’s words, the dopant “has to be such that it does not distort the original pure semiconductor lattice”, and a necessary condition is that the sizes of the dopant and the semiconductor atoms “should be nearly the same” (NCERT page 329).
• Si and Ge belong to the fourth group of the periodic table (group 14 today), so dopants are chosen from the nearby fifth and third groups (NCERT page 330):
  • pentavalent (valency 5): arsenic (As), antimony (Sb), phosphorus (P);
  • trivalent (valency 3): indium (In), boron (B), aluminium (Al).

n-type semiconductor and donors

• When a pentavalent atom takes the place of a Si atom, four of its five valence electrons form bonds with the four silicon neighbours. The fifth is only very weakly bound to its parent atom, because it sees the four bonding electrons as part of the atom’s effective core (NCERT page 330, NCERT Figure 14.7(a)).
• So very little energy sets the fifth electron free: about 0.01 eV in germanium and 0.05 eV in silicon, against about 0.72 eV and 1.1 eV to jump the forbidden band. Even at room temperature it is free to move through the lattice (NCERT page 330).
• The pentavalent atom donates an electron for conduction, so it is a donor impurity. The number of electrons the donors supply depends strongly on the doping level and, since nearly all donors are already ionised at room temperature, hardly grows as the crystal warms further; NCERT says “independent of any increase in ambient temperature” (page 330).
• NCERT slip (page 330): NCERT adds that the number of electron-hole pairs generated by the Si atoms themselves “increases weakly with temperature”. The equilibrium intrinsic carrier concentration rises steeply: NCERT Chapter 3 (page 91) explains that the number of carriers in a semiconductor grows so fast with temperature that its resistivity falls as it warms, and the current in an intrinsic semiconductor is highly temperature dependent (Wikipedia, “Intrinsic semiconductor”).
• The conduction electrons come from the donors and from intrinsic generation; the holes come only from intrinsic generation. With so many more electrons, holes recombine faster, so the number of holes falls even below its intrinsic value (NCERT page 330).
• Electrons are the majority carriers and holes the minority carriers: this is an n-type semiconductor (NCERT Eq. (14.3), page 331):

• NCERT Figure 14.7(b) draws n-type material as fixed donor cores, each with one extra positive charge, and their extra electrons (NCERT page 330).

p-type semiconductor and acceptors

• A trivalent atom such as Al, B or In has one valence electron fewer than Si. It forms covalent bonds with three Si neighbours but has no electron to offer the fourth, so that bond has a vacancy, a hole (NCERT page 331, NCERT Figure 14.8(a)).
• An electron from a neighbouring atom may jump into this vacancy, leaving a hole at its own site, so the hole is available for conduction (NCERT page 331).
• Once it has taken this fourth electron, the trivalent atom is effectively negatively charged: NCERT Figure 14.8(b) draws it as a core of one negative charge with its hole. One acceptor gives one hole, so the dopant is an acceptor (NCERT page 331).
• The holes from the acceptors add to the intrinsic holes, while the only source of conduction electrons is intrinsic generation; recombination cuts the electrons below n_i. Holes are the majority carriers and electrons the minority carriers: this is a p-type semiconductor (NCERT Eq. (14.4), page 331):

• The crystal stays neutral. The charge of the extra carriers is equal and opposite to that of the ionised dopant cores fixed in the lattice (NCERT page 331).
• With so many majority carriers about, the thermally produced minority carriers are more likely to meet one and be destroyed. So by adding carriers of one kind, the dopant indirectly lowers the number of the other kind (NCERT page 331).

Energy bands of doped semiconductors

• Doping adds energy states: donor levels E_D and acceptor levels E_A (NCERT page 331).
• In n-type Si the donor level E_D lies slightly below the bottom E_C of the conduction band, so a very small supply of energy lifts its electrons into the conduction band. At room temperature most donor atoms are ionised but very few Si atoms are, so most conduction electrons come from the donors (NCERT page 331, NCERT Figure 14.9(a)).
• NCERT Figure 14.9(a), which the text uses for n-type Si, marks the donor level about 0.01 eV below E_C, the germanium value; for silicon NCERT’s text gives 0.05 eV (page 330).
• NCERT gives the number of ionised Si atoms as “(~10¹²)” (page 331), with no unit or volume. The bare number cannot be a concentration without units. For comparison, in pure Si, NCERT Example 14.2’s 1.5 × 10¹⁶ pairs among 5 × 10²⁸ atoms m⁻³ gives about one conduction electron per 3.3 × 10¹² atoms. In the doped example the equilibrium hole density is only 4.5 × 10⁹ m⁻³. These are mobile-carrier population ratios, not a permanent census of ionised Si atoms.
• In p-type Si the acceptor level E_A lies slightly above the top E_V of the valence band. A very small supply of energy lets an electron from the valence band jump to E_A, ionising the acceptor; equivalently, the hole sinks from E_A into the valence band. At room temperature most acceptors are ionised, so the holes in the valence band come mainly from the impurity (NCERT page 332, NCERT Figure 14.9(b)).
• In thermal equilibrium in the nondegenerate regime, in intrinsic, n-type and p-type material alike, the electron and hole concentrations obey (NCERT Eq. (14.5), page 332; NCERT Summary point 9, page 340):

• NCERT calls this band picture “grossly approximate and hypothetical”, but useful for seeing how metals, insulators and semiconductors differ (NCERT page 332).
• Conductivity (beyond the NCERT text): as in pure silicon, σ = e(n_eμ_e + n_hμ_h), where μ_e and μ_h are the mobilities. In n-type material the holes add almost nothing, so σ ≈ en_eμ_e. See Example 3.

The numerical examples assume thermal equilibrium, moderate nondegenerate doping, full shallow-dopant ionisation and negligible compensation unless stated. At very low temperatures dopants can freeze out; at high temperatures intrinsic carriers can dominate.

Solved examples

Common mistakes to avoid

Quick revision summary

• Doping a few ppm of a suitable impurity raises a semiconductor’s conductivity manifold; the dopant must not distort the lattice, so its atoms must be nearly the same size as the host atoms.
• Pentavalent donors (As, Sb, P) give n-type: electrons are the majority carriers, n_e ≫ n_h.
• Trivalent acceptors (B, Al, In) give p-type: holes are the majority carriers, n_h ≫ n_e.
• Donor levels lie just below E_C, acceptor levels just above E_V; at room temperature most dopants are ionised.
• In nondegenerate thermal equilibrium n_en_h = n_i²; the crystal is electrically neutral.

Questions students ask

What is the difference between n-type and p-type semiconductors?

An n-type semiconductor is silicon or germanium doped with a pentavalent donor, so electrons are the majority carriers and the fixed donor ions are positive, while a p-type semiconductor is doped with a trivalent acceptor, so holes are the majority carriers and the fixed acceptor ions are negative (NCERT pages 330–331). Their extra energy levels sit at opposite edges of the gap: the donor level just below the conduction band, the acceptor level just above the valence band (NCERT pages 331–332). Beyond the NCERT text, for the same doping level n-type silicon conducts better, because electrons are roughly three times as mobile as holes (Wikipedia, “Electron mobility”).

Why is doping necessary in semiconductors?

Doping is necessary because a pure semiconductor’s conductivity at room temperature is too low for any important electronic device, while a few parts per million of a suitable impurity raises it many times (NCERT page 329). Beyond the NCERT text, undoped (intrinsic) layers still have uses, as in p-i-n diodes for light and radiation detectors (Wikipedia, “Diode”). Doping also chooses the majority carrier, electrons in n-type or holes in p-type, and joining the two makes a p-n junction. Once nearly all donors are ionised, as at room temperature, the electrons they supply hardly change with temperature (NCERT pages 330–331), while a pure crystal’s carriers multiply as it warms (reasoned).

What is the mass action law in semiconductors, and why does it hold?

The mass action law in semiconductors says that in nondegenerate thermal equilibrium the product of the electron and hole concentrations is fixed at a given temperature, n_en_h = n_i², in pure, n-type and p-type material alike (NCERT Eq. (14.5), page 332). In the nondegenerate band model, n_e ∝ exp[(E_F − E_C)/kT] and n_h ∝ exp[(E_V − E_F)/kT]; multiplying cancels the common equilibrium Fermi energy E_F. This does not assume every recombination mechanism is simply proportional to np. Strong degeneracy or nonequilibrium illumination needs a more general treatment (Wikipedia, “Charge carrier density” and “Carrier generation and recombination”). So silicon with n_e = 3 × 10^²¹ m^⁻³ and n_i = 1.5 × 10^¹⁶ m^⁻³ has n_h = 7.5 × 10^¹⁰ m^⁻³.

What is doping concentration, and how is it calculated?

Doping concentration is the number of dopant atoms per unit volume of the crystal, N_D for donors or N_A for acceptors, in m^⁻³ (books that use cm^⁻³ give numbers 10^⁶ times smaller); it is the fraction of host atoms replaced times the number of host atoms per unit volume, as in NCERT Example 14.2 (page 332). For silicon, with 5 × 10^²⁸ atoms m^⁻³, doping with one phosphorus atom in 10^⁷ gives N_D = 5 × 10^²¹ m^⁻³, and at room temperature most of these donors are ionised (NCERT page 331). “Light” and “heavy” doping are relative to the material and application; there is no universal ppm boundary.

Why are pentavalent and trivalent impurities commonly used to dope silicon?

Pentavalent and trivalent impurities are used because each has one valence electron more or fewer than silicon, so each atom supplies one loosely bound electron or hole that is freed at room temperature (NCERT pages 330–332). Beyond the NCERT text, other impurities such as gold make levels deep in the band gap that capture carriers rather than supply them, and gold is added to some silicon devices to shorten carrier lifetimes (Wikipedia, “Doping (semiconductor)”). In compound semiconductors a foreign atom is not the only way: native defects and departures from stoichiometry can also affect carriers. NCERT’s Ga-rich/As-rich illustration (Points to ponder 2, page 341) is qualitative, not a recipe for a homogeneous Ga_1.1As_0.9 crystal.

Practice questions

Try each question first, then open the answer to check your working.

1. Q1. Why is a pentavalent impurity in silicon called a donor?

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   Four of its five valence electrons form bonds; the fifth is so weakly bound that it is freed at room temperature, so the atom donates an electron for conduction (NCERT page 330).

2. Q2. Where does the acceptor level lie in the band diagram of p-type silicon, and what happens at room temperature?

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   Slightly above the top of the valence band, E_V. Valence electrons easily jump to it, so most acceptors are ionised and holes are left in the valence band (NCERT page 332).

3. Q3. A Si sample is doped so that n_e = 10²² m⁻³. With n_i = 1.5 × 10¹⁶ m⁻³, find n_h.

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   n_h = n_i²/n_e = (2.25 × 10³²)/10²² = 2.25 × 10¹⁰ m⁻³ (NCERT Eq. (14.5), page 332).